How to use Blender's dynamic sky add-on?
First go here: http://www.dragoneex.com/downloads/dynamic-skyadd-on and look at "Instruction" section. This is self-explanatory and will help with further tasks.

Controlling sun/moon

These sphere you are referring to is nothing more then Vector > Normal Node. There is a little info in docs about it so I'll quickly show you how to control sun with it in most understandable way.



Simply, go to the Top View and this is the way you can relate to the sphere shading. If sphere is lit from the left, mesh will be lit the same way. If light will be visible at the center of the sphere, mesh will be lit from the top.

Daytime sky without clouds

This one is the easiest one. You only need to change Cloud Opacity to 0. These will hide our clouds. You can further adjust colors, brightness, etc.

Used colors


Sky - #b0ecff
Horizon - #fbeaea
Cloud - #ffffff
Sun - #bcbcbc




Sunset sky with clouds

In this example colors are crucial. Darker sky, orange horizon, warmer sun light, lower Brightness. Also you will need to tweak Clouds Opacity and Density as needed. Don't forget to place sun near Horizon using tip from first paragraph.

Used colors


Sky - #385179
Horizon - #fb8648
Cloud - #ffb9a8
Sun - #bc9481




Night-time sky with stars

This is where the fun begins and in this paragraph you will learn how to push this add-on to the limits. It's not the easiest way but it will give you total control over it. One thing I can't solve is moon opacity over stars. Maybe someone could help here.

Used colors


Sky - #151f2e
Horizon - #191825
Cloud - #ffffff
Sun - #7688a0


First as before adjust overall mood and colors. Brightness down, colors darker, cool sun/moon light. And after it...



...go to the Node Editor and switch to World. This is an overview of what part of those noodles control what we can see.



Stars



To make stars from clouds change bottom Noise Texture to Voronoi Texture, set Scale in Mapping node to 1 (X, Y, Z) and adjust Color Ramp (from Voronoi) to have "sharp" stars. Upper Noise Texture is controlled by Clouds Density value in add-on tab. This is basically "mask" for the stars/clouds. If set to 1 it will create "gaps" between them.

Moon/Sun



In this part you can control how big sun/moon will be, and how big will be its halo. Nothing really to explain here. Slide Color Ramp values and change upper Multiply Node value.

Horizon level



Another fairly easy setup, left values from Color Ramps are controlling horizon/sky/clouds height. Right (middle) ones are controlling sharpness.

Sun color

You can change Sun color in MixRGB node showed in screenshot. You can also change blending mode to Linear Light to have stronger colors.



Final thoughts

As far as last method isn't convenient one and feels more like "hacky" way to do this I think it could give you pretty much anything you need from this add-on. My advice would be - play with values, nodes, colors in Node Editor to find new ways to use it. 

In a matter of making it realistic. Well. Overall lightning is ok. I don't know if it's physically correct but it's really good. One thing that's not realistic are clouds and stars and there is probably nothing you can do about it in this add-on. Of course you can create amazing procedural cloud setup and use it in a place of two Noise Textures but still they will be just 2D representation of clouds without Volume Scattering and such.

If something is not clear in my answer, please point it in the comments. It was pretty long one and something could be missed.